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Chapter 15 Multiple Integrals 15-2 & 15-3 Iterated Integrals 15-1, 15-2, & 15-3 Double Integrals and Volume 15-4 Double Integrals in Polar Coordinates 15-6 Surface Area (4th ed.) 15-6 Triple Integrals 15-7 Triple Integrals in Cylindrical Coordinates 15-8 Triple Integrals in Spherical Coordinates
The following notes are for the Calculus D (SDSU Math 252) classes I teach at Torrey Pines High School. I wrote and modified these notes over several semesters. The explanations are my own; however, I borrowed several examples and diagrams from the textbooks* my classes used while I taught the course. Over time, I have changed some examples and have forgotten which ones came from which sources. Also, I have chosen to keep the notes in my own handwriting rather than type to maintain their informality and to avoid the tedious task of typing so many formulas, equations, and diagrams. These notes are free for use by my current and former students. If other calculus students and teachers find these notes useful, I would be happy to know that my work was helpful. - Abby Brown SDUHSD Calculus III/D SDSU Math 252 Abby Brown www.abbymath.com San Diego, CA
*Calculus: Early Transcendentals, 6th & 4th editions, James Stewart, ©2007 & 1999 Brooks/Cole Publishing Company, ISBN 0-495-01166-5 & 0-534-36298-2. (Chapter, section, page, and formula numbers refer to the 6th edition of this text.) *Calculus, 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards, ©1994 D. C. Heath and Company, ISBN 0-669-35335-3.
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x
y4
x
2y
1
1
2
How do we decide dxdy or dydx? Consider both (1) the shape of the region and (2) the integrand. Usually one order of integration is easier than the other. To switch the order of integration, sketch the region determined by the limits and use the graph to help you write new limits.
3
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Given a surface f(x,y), we can approximate the volume under the surface over a given region R in the xy-plane. Break up the region into squares, calculate the height at a corresponding point in each square, calculate the volume of each rectangular prism, and add all of the volumes together. Then change the approximation to an infinite number of prisms by taking the limit.
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